Why does a bump/normal map change the look of a plane to which they're applied to, even if the plane only has 1 normal?
A normal can be thought of as the direction a face is pointing.  The face (usually) has one normal, it's just a vector pointing into space going outwards from the face.  For our purposes here, it defines the average angle of reflection of light rays.
When you modify it with a Normal Map, you're giving it additional instructions on how to handle light across the face.  For every evaluated pixel, the light is taking the face normal and your Normal Map information into account.
So it's not that it is creating more normals.  There is only one, with a little math being done for every pixel sampled.  Very easy for your processor/gpu.
From Wikipedia

To calculate the Lambertian (diffuse) lighting of a surface, the unit
vector from the shading point to the light source is dotted with the
unit vector normal to that surface, and the result is the intensity of
the light on that surface. Imagine a polygonal model of a sphere - you
can only approximate the shape of the surface. By using a 3-channel
bitmap textured across the model, more detailed normal vector
information can be encoded. Each channel in the bitmap corresponds to
a spatial dimension (X, Y and Z). This adds much more detail to the
surface of a model, especially in conjunction with advanced lighting
techniques.